How can a huge-scale chemical process be managed to guarantee that the largest possible yield of a certain substance is produced? How can we ensure that buildings and other infrastructure are optimally designed? How can fluid dynamics processes be impacted so as to minimize turbulence or maximize flow in a particular region? These, and many other important questions from science, engineering, and industry, may be tackled through the optimization and control of problems involving partial differential equations (PDEs). PDEs are used to describe mathematically how real-world physical systems behave: they can model cell biology, chemical reactions, processes in mathematical finance, fluid flow, quantum mechanics, and a vast range of other processes. What we are particularly interested in is the optimization of such problems, where we apply some external forces on the dynamics so that the system will behave in the 'best' possible way. This motivates the main focus of this project: the study of PDE-constrained optimization, including particular problem formulations which are often referred to as infinite-horizon control or model-predictive control problems. The possibilities such formalisms offer is enormous, driving cutting-edge research in engineering, systems biology, chemical processes, imaging, and many other fields. Whereas many such problems can be clearly stated on paper, accurately resolving them on a computer is a very important, and difficult, challenge. Indeed, for many problems with information provided at a very fine level, in particular resulting from processes driven by vast quantities of data, the resulting systems of equations are of such enormous scale that producing accurate numerical solutions can be intractable. This work seeks to resolve this challenge, by bringing to bear modern technologies from the field of numerical linear algebra, in particular through the timely and exciting research area of randomized linear algebra. The exploitation of current methodologies can ensure the generation of robust solutions in real-time, while minimizing computer storage requirements, and often enabling the use of parallel computing. The usage of randomization within solvers for PDEs themselves has been well established, however the development of such solvers is so far an underexplored area for optimization and control problems where the PDEs act as constraints. We will meet this outstanding challenge through four ambitious work packages: (i) using randomization in eigenvalue iteration for parabolic PDEs, an important class of PDEs which describe diffusion-driven processes in particular; (ii) randomized features within iterative methods for modelling processes with nonlinear phenomena; (iii) randomization within model-order reduction for PDE-constrained optimization, where the computational complexity of the model itself is reduced to make feasible a range of numerical algorithms for the solution; (iv) the use of randomized solvers for problems which have uncertain inputs. The final package will bring together all of the previous work, devising an overarching framework for treating optimization problems with randomness built in. Our new algorithms will be analysed theoretically and validated numerically, on a wide variety of huge-scale problems. Interaction with industry and across academic disciplines is a key outcome. Industry impact will be generated in collaboration with project partners FESTO and Arup, with whom we will apply our methodology to optimization and control problems that have a key link to national economic challenges. We will release code libraries, and organise a workshop with academic and industrial invitees, to further enhance the scientific and commercial impact of these new developments.

The Digital Innovation and Circular Economy (DICE) Network+ aims to drive a transformative shift in the sustainability and circularity of digital and communication technologies. Our vision leverages the digital revolution to foster a circular economy across sectors and value chains, adopting a "network of networks" approach for interdisciplinary collaboration, research, and technological innovation. DICE focuses on overcoming challenges such as the lack of circular economy principles in digital technology design and manufacture, and the poor understanding and coordination of digital advancements in supporting the transition towards a UK circular economy. Our network comprises 11 investigators, from engineering, materials science and social sciences and a wide range of partners, including universities, industry stakeholders, and public bodies. It aims to benefit stakeholders through the co-creation of innovative solutions, fostering knowledge exchange, supporting projects that promote digitally enabled circular economy adoption and guidance on future policy making and industrial decision making. The approach centres around interdisciplinary collaboration, leveraging our extensive existing networks (over £160m of funding since 2020) for maximum impact, and a structured programme of network engagement under the four pillars of Insight and Evidence, Inclusive Community, Capacity Building and Knowledge Exchange, and Research Impact and Legacy. DICE's activities include mapping exercises, webinars, annual showcases, co-creation workshops, knowledge exchange placements, feasibility studies, and demonstrator projects, culminating in the development of a 10-year vision and roadmap towards a digitally enabled CE to guide future policy making, industrial decision making, investment and technological development.